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Βραχυστόχρονη Καμπύλη
Βραχυστόχρονος Καμπύλη brachistochrone Curve thumb|300px| [[Βραχυστόχρονη Καμπύλη ]] thumb|300px| [[Μαθηματικά Γεωμετρία Καμπύλη Καμπύλες ---- ---- (a) Δεξιά Στροφοειδής Καμπύλη (Right strophoid) ---- (b) Τρίαινα Newton (Trident of Newton) ---- © Καρδιοειδής Καμπύλη (Cardioid) ---- (d) Δελτοειδής Καμπύλη (Deltoid) ---- (e) Δαιμονική Καμπύλη (Devil on two sticks) ----- (f) Λημνίσκος Bernoulli (Lemniscate of Bernoulli) ---- (g) Επιτροχοειδής Καμπύλη (Epitrochoid) ---- (h) Ροδονέα Καμπύλη (Rhodonea curve) ---- (i) Καμπύλη Bowditch (Bowditch curve) ---- (j) Σπείρα Fermat (Fermat's spiral) ---- (k) Λογαριθμική Σπείρα (Logarithmic spiral) ---- (l) Κυκλοειδής Καμπύλη (Cycloid) ]] thumb|300px| [[Βραχυστόχρονη Καμπύλη ]] thumb|thumb|300px| [[Βραχυστόχρονη Καμπύλη ]] - Είδος καμπύλης. Ετυμολογία Η ονομασία "Βραχυστόχρονη" σχετίζεται ετυμολογικά με την λέξη "χρόνος". Εισαγωγή Το ερώτημα που τίθεται είναι: ποια καμπύλη αντιστοιχεί στο βραχύτερο χρονικό διάστημα, δηλαδή ποια καμπύλη είναι η βραχυστόχρονη; Ο πρώτος που έθεσε το πρόβλημα της βραχυστόχρονης ήταν ο Johann Bernoulli, στην ‘πρόσκληση’ του οποίου ανταποκρίθηκαν ο Leibniz, ο Jacob Bernoulli, ο L'Hospital, ο Νεύτων κ.α. Όλοι αυτοί οι μαθηματικοί κατέληξαν στο ίδιο συμπέρασμα: Η βραχυστόχρονη είναι η Κυκλοειδής Καμπύλη. (Η κυκλοειδής είναι η τροχιά που διαγράφει κάποιο συγκεκριμένο σημείο ενός κύκλου που κυλίεται, χωρίς να ολισθαίνει, σε μια ευθεία γραμμή) The brachistochrone is the cycloid Given two points A'' and ''B, with A'' not lower than ''B, only one "upside down" cycloid, i.e., with a horizontal base and the cusps pointing upward, passes through both points, has a cusp at A'', and has no maximum points between ''A and B'': the brachistochrone curve. It is a segment of a cycloid arch including at most two cusps, one at ''A, and possibly another one at B'' in the special case that ''B is at the same height as A''. Unlike the full cycloid, for which the tangent line is undefined at the cusps, the brachistochrone curve has a vertical tangent line at these cusps. The curve does not depend on the body's mass or on the strength of the gravitational constant. The problem can be solved with the tools from the calculus of variations and optimal control.Ross, I. M. The Brachistochrone Paridgm, in ''A Primer on Pontryagin's Principle in Optimal Control, Collegiate Publishers, 2009. ISBN 978-0-9843571-0-9. If the body is given an initial velocity at A'', or if friction is taken into account, then the curve that minimizes time will differ from the one described above. Johann Bernoulli's solution According to Fermat’s principle: ::''The actual path between two points taken by a beam of light is the one which is traversed in the least time. In 1697 Johann Bernoulli used this principle to derive the brachistochrone curve by considering the trajectory of a beam of light in a medium where the speed of light increases following a constant vertical acceleration (that of gravity g''). Conservation of energy can be used to express the speed of a body in a constant gravitational field as: : v=\sqrt{2gy} , ::where ''y represents the vertical distance the body has fallen. The speed of motion of the body along an arbitrary curve does not depend on the horizontal displacement. Johann Bernoulli noted that the law of refraction gives a constant of the motion for a beam of light in a medium of variable density: : \frac{\sin{\theta}}{v}=\frac{1}{v}\frac{dx}{ds}=\frac{1}{v_m} , ::where vm is the constant and '' \theta '' represents the angle of the trajectory with respect to the vertical. The equations above allow us to draw two conclusions: # At the onset, the angle must be zero when the particle speed is zero. Hence, the brachistochrone curve is tangent to the vertical at the origin. # The speed reaches a maximum value when the trajectory becomes horizontal and the angle θ = 90°. Simplifyingly assuming that the particle (or the beam) with coordinates (x,y) departs from the point (0,0) and reaches maximum speed after a falling a vertical distance D'': : v_m=\sqrt{2gD} . Rearranging terms in the law of refraction and squaring gives: : v_m^2 dx^2=v^2 ds^2=v^2 (dx^2+dy^2) which can be solved for ''dx in terms of dy: : dx=\frac{v\, dy}{\sqrt{v_m^2-v^2}} . Substituting from the expressions for v'' and ''vm above gives: : dx=\sqrt{\frac{y}{D-y}}dy which is the differential equation of an inverted cycloid generated by a circle of diameter D''. Jakob Bernoulli's solution Johann's brother Jakob showed how 2nd differentials can be used to obtain the condition for least time. A modernized version of the proof is as follows. If we make a negligible deviation from the path of least time, then, for the differential triangle formed by the displacement along the path and the horizontal and vertical displacements, : ds^2=dx^2+dy^2 . On differentiation with ''dy fixed we get, : 2ds\ d^2s=2dx\ d^2x . And finally rearranging terms gives, : \frac{dx}{ds}d^2x=d^2s=v\ d^2t where the last part is the displacement for given change in time for 2nd differentials. Now consider the changes along the two neighboring paths in the figure below for which the horizontal separation between paths along the central line is d2x (the same for both the upper and lower differential triangles). Along the old and new paths, the parts that differ are, : d^2t_1=\frac{1}{v_1}\frac{dx_1}{ds_1}d^2x : d^2t_2=\frac{1}{v_2}\frac{dx_2}{ds_2}d^2x For the path of least times these times are equal so for their difference we get, : d^2t_2-d^2t_1=0=\bigg(\frac{1}{v_2}\frac{dx_2}{ds_2}-\frac{1}{v_1}\frac{dx_1}{ds_1}\bigg)d^2x And the condition for least time is, : \frac{1}{v_2}\frac{dx_2}{ds_2}=\frac{1}{v_1}\frac{dx_1}{ds_1} thumb|center|500px|[[Βραχυστόχρονη Καμπύλη.]] Υποσημειώσεις Εσωτερική Αρθρογραφία * Αρχή Ελαχίστου Χρόνου * Κυκλοειδής Καμπύλη, * Καρδιοειδής Καμπύλη, * Νεφροειδής Καμπύλη, * Δελτοειδής Καμπύλη, * Αστεροειδής Καμπύλη, * Ελικοειδής Καμπύλη, * Κογχοειδής Καμπύλη, Βιβλιογραφία * * Ιστογραφία * Ομώνυμο άρθρο στην Βικιπαίδεια * Ομώνυμο άρθρο στην Livepedia * Brachistochrone is Cycloid, proofwiki.org * Η κυκλοειδής καμπύλη είναι «ισόχρονη» και «βραχυστόχρονη» *whistleralley.com * arxiv.org * Variational Methods * videoclip Category: Καμπύλες